This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1865 Excerpt: ...one of the arbitrary constants av be presented in the form Differentiating with respect to each of the independent variables we have a system of n equations of the forms These n equations enable us first to eliminate the n--1 constants a2, an, and so deduce the partial differential equation sought in the form secondly ...
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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1865 Excerpt: ...one of the arbitrary constants av be presented in the form Differentiating with respect to each of the independent variables we have a system of n equations of the forms These n equations enable us first to eliminate the n--1 constants a2, an, and so deduce the partial differential equation sought in the form secondly to determine the n--1 constants as functions of xl, ... xK, z, pv... pn in the forms As the system formed of these n--1 equations, together with the previous one, is merely another form of the system (2) obtained by directly differentiating the primitive, it follows that if from these equations we deduce the values ofp ... pn as functions of x0... xn, z2, ... an, and substitute them in the equation they will render that equation integrable, and its integral will be the complete primitive (1), the constant a1 being regained by integration. B.D.E. II. 7 Examining the system (3), (4) we see that the first members of all the equations which it contains are functions of x1, ... xn, z, p1, ...pn, while the second members are constants. The question then arises, What mutual connexion exists among these functions in virtue of which they yield values of p ...pn, which render the equation (5) integrable? The answer to this question must involve the entire theory of the solution of partial differential equations of the first order, so far as relates to the determination of a complete primitive. Given a partial differential equation of the form (3) it is evident that if we can construct a system of associated equations (4) possessing the character above described, the final value of z obtained by integration of (5) will both satisfy the given equation and contain the requisite number of arbitrary constants. It does not follow from this that it wi
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